Seamus Coleman gave the visitors the lead in the 66th minute at the Liberty Stadium, before Bryan Oviedo inadvertently diverted a Dwight Tiendalli effort into his own goal four minutes later.
The game was settled by a superb Barkley free-kick six minutes from time and Martinez confessed he was running out of words to describe the bright English prospect after watching his side move up to fourth in the Premier League.
"Whatever you say about him, you always end up with an understatement," he said. "He is a quite incredible diamond of English football.
"He's an incredible listener - a young man with an incredible potential.
"Every game he comes up with something different and every game he shows you that he can develop.
"He's a real natural talent, and everything he does technically is as good as it gets.
"Right, left, great balance. We didn't know he could take free-kicks, but it's good to know from now on."
Despite the fact Everton have lost just once in the Premier League so far this season, Martinez is refusing to get carried away with the plaudits they are receiving.
"It would be a mistake for us to talk about the final aim for the season in December," he continued. "I always said that all we want to do is try to fulfil our potential.
"I still don't think we have fulfilled our maximum and that's what we're focusing on.
"When you get to April it's more realistic to talk about what sort of aims you can fight for.
"It's an open competition, but from our point of view, it's pointless to look at the table now.
"I think we need to bring two players in in January and make sure that we keep working and developing to fulfil our potential."
